Background to Gottlob Frege
Gottlob Frege (1848–1925) Life’s work: logicism (the reduction of arithmetic to logic). This entailed: •Inventing (discovering?) modern logic, including quantification, variables, etc. •Investigating the properties of the language in which this logic was spelled out.
Step 1: Begriffsschrift (1879) …in which Frege comes up with "classical" logic.
What is Logic? Deductive logic is the attempt to formulate a precise theory that predicts which conclusions follow from which premises. These predictions are based solely on the structures of the sentences involved.
Syllogistic Logic From Aristotle until the 19th Century, almost all logic was syllogistic. E.g.: All men are mortals. All A are B. All mortals will die. All B are C. So, all men will die. So, all A are C. Syllogistic logic assumes that sentences are structured into subjects and predicates. Frege showed that this is a bad way to do things.
The Problem with Subject-Predicate Structure It can’t help us to predict certain good inferences. E.g.: John loves Steve. John loves Steve. So John loves someone. So Steve is loved by someone.
Frege on Sentence Structure Frege pointed out that there are multiple ways to break sentences down into parts: John loves ___ + Steve ____ loves Steve + John ____ loves _ _ _ _ + John + Steve The “unsaturated” parts are predicates. They refer to “concepts” ( Begriffe ). The “saturated” parts (names) refer to “objects”. The “extension” of a concept is the set of all the things that the concept is true of.
Frege on Sentence Structure Frege pointed out that there are multiple ways to break sentences down into parts: John loves ___ + Steve ____ loves Steve + John ____ loves _ _ _ _ + John + Steve The “unsaturated” parts are predicates. They refer to “concepts” ( Begriffe ). The “saturated” parts (names) refer to “objects”. Thus the name that Frege gave to his logic: “concept script” ( Begriffsschrift )
Functions and Arguments Frege argued that we should think of concepts as functions (in the mathematical sense). They take objects as their arguments and map the to truth values (truth or falsity). John loves ___(Steve) = truth John loves ___(Donald) = falsity
Quantifiers What about the structure of a quantificational sentence, like these? John loves everyone Someone loves Steve Everyone loves someone Quantifier words, like “everyone” and “someone”, don’t look like they refer to individual objects. Frege argued that they refer to second-level concepts. Concepts that take first-level concepts as arguments and map them to truth values: everyone( ____ loves Steve ) (the concept of being being true of everyone applies to the concept of loving Steve)
Notation English: Everything is good. The notation I’ve been using: everything( ____ is good ) a Frege’s Notation: G(a) or: ( ∀ x)Gx (x)Gx Contemporary logical notation:
Notation English: Everything good is not bad. a Frege’s Notation: B(a) G(a) ( ∀ x)(Gx → ~Bx) Contemporary logical notation:
Step 2: Die Grundlagen der Arithmetic (1884) …in which Frege proposes his definition of number.
0 1 0 is the set of all sets 1 is the set of all sets containing no members containing a single member 0 = df { s : ¬( ∃ x )( x ∈ s )} 1 = df { s : ( ∃ x )( x ∈ s & ( ∀ y )( y ∈ s ⊃ y=x ))}
3 2 3 is the set of all sets containing 2 is the set of all sets that have exactly three members exactly two members = df { s : ( ∃ x )( ∃ y )( ∃ z )( x ∈ s & y ∈ s & = df { s : ( ∃ x )( ∃ y )( x ∈ s & y ∈ s & x ≠ y z ∈ s & x ≠ y & x ≠ z & y ≠ z & ( ∀ w ) & ( ∀ z )( z ∈ s ⊃ (z ≠ x ⊃ z=y ))} ( w ∈ s ⊃ ( w ≠ x ⊃ ( w ≠ y ⊃ w=z ))))}
Step 3: Die Grundgesetze der Arithmetic (1893–1903) …in which Frege attempts to derive arithmetic from logic
Ti e number that belongs to the concept F = the number that belongs to the concept G i ff Ti ere is a one-to-one correspondence between the Fs and the Gs.
0 = the extension of the concept: is equinumerous with the concept: is not self-identical 0 = {s : s is equinumerous with { x : x ≠ x }}
1 = the extension of the concept: is equinumerous with the concept: is identical to 0 1 = {s : s is equinumerous with 0} 1 = {s : s is equinumerous with {{ x : x ≠ x }} }
2 = the extension of the concept: is equinumerous with the concept: is identical to 0 or is identical to 1 2 = {s : s is equinumerous with { x : x =0 ∨ x =1}} 2 = {s : s is equinumerous with {0, 1}
PEANO POSTULATES (1) 0 is a number (2) Ti e successor of any number is a number (3) No two numbers have the same successor (4) 0 is not the successor of any number (5) Any property which belongs to 0, and also to the successor of every number which has the property, belongs to all numbers.
FREGE’S BASIC LAW 5 For any concepts, F and G , the extension of F is identical to the extension of G if and only if for every object a , Fa if and only if Ga . A CONSEQUENCE/PRESUPPOSITION Every concept F has an extension.
Step 2.5: Three Philosophical Papers Funktion und Begriff (1891) (‘Function and Concept’) …in which Frege proposes that concepts (the meanings of predicates) should be thought of as functions . Über Begriff und Gegenstand (1892) (‘Concept and Object’) …in which Frege argues that concepts/functions differ from objects in that they’re "unsaturated". Über Sinn und Bedeutung (1892) (‘Sense and Reference’) …in which Frege elaborates his theory of words’ meanings.
Über Sinn und Bedeutung (‘Sense and Reference’) The Problem: “ π ” and “3.14159265359…” refer to the same object I.e., these two symbols have the same meaning. So then why do we learn something when we find out that π = 3.14159265359… After all, we can’t learn anything from π = π But we’ve only substituted synonyms for synonyms here.
"It is natural, now, to think of there being connected with a sign (name, combination of words, letter), besides that to which the sign refers, which may be called the reference of the sign, also what I should like to call the sense of the sign, wherein the mode of presentation is contained."
Each expression can have (at least) the following two semantic properties: • sense ( Sinn ) • reference ( Bedeutung ) *(Frege also mentions a third kind in passing: coloring ( Färbung ).
Frege’s Puzzle: Compare: (1) Hesperus is identical to Hesperus. (2)Hesperus is identical to Phosphorus. and: (3) The morning star is identical to the morning star. (4) The morning star is identical to the evening star.
Frege’s Puzzle (in general): Identity statements of the form ‘ a = a ’ are trivial, but those of the form ‘ a = b ’ are nontrivial. We get information from the second kind but not the first kind. They differ in cognitive value . If the meaning of an expression is just its referent, we can’t explain this.
Frege’s Solution to his Puzzle: “If we found "a=a" and "a=b" to have different cognitive values, the explanation is that for the purpose of knowledge, the sense of the sentence, viz., the thought expressed by it, is no less relevant than its referent, i.e., its truth value. If now a=b, then indeed the referent of "b" is the same as that of "a," and hence the truth value of "a=b" is the same as that of "a=a." In spite of this, the sense of "b" may differ from that of "a," and thereby the sense expressed in "a=b" differs from that of "a=a." In that case the two sentences do not have the same cognitive value.”
Job description for sense: • It is responsible for the expression’s "cognitive value”—i.e., what you can learn from sentences containing the expression.
Frege’s Anti-Psychologism “The sense of a proper name is grasped by everybody who is sufficiently familiar with the language or totality of designations to which it belongs; but this serves to illuminate only a single aspect of the referent, supposing it to exist. Comprehensiveknowledge of the refer- ent would require us to be able to say immediately whether every given sense belongs to it. To such knowledge we never attain.” p.210–211
Frege’s Anti-Psychologism "The reference and sense of a sign are to be distinguished from the associated idea. … The idea is subjective: one man’s idea is not another. … This constitutes an essential difference between the idea and the sign’s sense, which may be the common property of many and therefore is not a part of a mode of the individual mind." (8th paragraph)
Frege’s Anti-Psychologism "By a thought I understand not the subjective performance of thinking but its objective content, which is capable of being the common property of several thinkers." (fn.7)
The Sense of a Sentence is a Thought "By a thought I understand not the subjective performance of thinking but its objective content, which is capable of being the common property of several thinkers." (fn.7)
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